Group Theory

Join Medium with my referral link - applied.math.coding Get access to all of my stories and to thousands more on Medium from other writers. As my strong believe is, Medium is…medium.com This story follows-up and concludes the series about basic aspects in group theory. One of the cornerstone results so far was the decomposition of an commutative, finitely generated group (see here). This led to a split into elements of finite order and elements of infinite order where we have seen…

Join Medium with my referral link - applied.math.coding Get access to all of my stories and to thousands more on Medium from other writers. As my strong believe is, Medium is…medium.com Now where we have seen the ability to decompose a commutative, finitely generated group into as small as possible non-trivial factors (see here), the natural question arises: Is this decomposition unique? This story aims to provide the necessary tools in order to show such a uniqueness result.

Join Medium with my referral link - applied.math.coding Get access to all of my stories and to thousands more on Medium from other writers. As my strong believe is, Medium is…medium.com In this story we follow-up what we have started here about looking into the so-called Sylow decomposition of a commutative, finite generated group with all elements having finite order. …

Join Medium with my referral link - applied.math.coding Get access to all of my stories and to thousands more on Medium from other writers. As my strong believe is, Medium is…medium.com In the previous story (see here) we have seen a quite remarkable result about decomposing a commutative, finitely generated group. This decomposition reveals some further concepts that you inevitable encounter in applications of groups of the aforementioned type.

It’s deeper than you might think. — If you walked by a church in England in the early 1600s and heard its bells ringing, you might have continued on your way thinking the bells were being rung in a random, or at least uninteresting, order. …

A Journey Through Its Basic Concepts and Applications — A rather large part of mathematics is fundamentally about relationships. Mathematicians spend a lot of time trying to find relationships between numbers, collections of numbers, collections of collection of numbers, circles, triangles, infinities, and the list goes on and on. Let’s examine a particularly useful type of relationship: mappings. Set…

Join Medium with my referral link - applied.math.coding Get access to all of my stories and to thousands more on Medium from other writers. As my strong believe is, Medium is…medium.com The last result of this series (see here) stated that the additive groups Z and Z_q for q being a power of a prime number, are not decompose-able into non-trivial direct sums. The ultimate goal of this story is a very important statement about decomposing a commutative, finitely generated group…

Join Medium with my referral link - applied.math.coding Get access to all of my stories and to thousands more on Medium from other writers. As my strong believe is, Medium is…medium.com In our previous story (see here) of this series, we have seen the order of an element always divides the order of the group — at least for finite groups. Now, let us apply this statement to the question if certain groups can be decomposed into non-trivial direct sums.

Join Medium with my referral link - applied.math.coding Get access to all of my stories and to thousands more on Medium from other writers. As my strong believe is, Medium is…medium.com Following-up what has been started by this series, we are going to consider in this story an important relation between the order of an element in a group and the order of the group itself. Remember, the order of a group G is just its cardinality |G|. The order of…